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Theorem ereldm 8794
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1 (𝜑𝑅 Er 𝑋)
ereldm.2 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Assertion
Ref Expression
ereldm (𝜑 → (𝐴𝑋𝐵𝑋))

Proof of Theorem ereldm
StepHypRef Expression
1 ereldm.2 . . . 4 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21neeq1d 2998 . . 3 (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅))
3 ecdmn0 8793 . . 3 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
4 ecdmn0 8793 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
52, 3, 43bitr4g 314 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
6 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
7 erdm 8754 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
86, 7syl 17 . . 3 (𝜑 → dom 𝑅 = 𝑋)
98eleq2d 2825 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
108eleq2d 2825 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
115, 9, 103bitr3d 309 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wne 2938  c0 4339  dom cdm 5689   Er wer 8741  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-er 8744  df-ec 8746
This theorem is referenced by:  erth  8795  brecop  8849  eceqoveq  8861
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